# Properties

 Label 1568.2.a.x Level $1568$ Weight $2$ Character orbit 1568.a Self dual yes Analytic conductor $12.521$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1568.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.5205430369$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{5})$$ Defining polynomial: $$x^{4} - 6 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} -2 \beta_{1} q^{5} + 7 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} -2 \beta_{1} q^{5} + 7 q^{9} -\beta_{3} q^{11} + 2 \beta_{3} q^{15} -3 \beta_{1} q^{17} -\beta_{2} q^{19} -2 \beta_{3} q^{23} + 3 q^{25} -4 \beta_{2} q^{27} -6 q^{29} -2 \beta_{2} q^{31} + 10 \beta_{1} q^{33} + 2 q^{37} -\beta_{1} q^{41} + \beta_{3} q^{43} -14 \beta_{1} q^{45} + 2 \beta_{2} q^{47} + 3 \beta_{3} q^{51} + 6 q^{53} + 4 \beta_{2} q^{55} + 10 q^{57} -\beta_{2} q^{59} + 6 \beta_{1} q^{61} + 20 \beta_{1} q^{69} -2 \beta_{3} q^{71} + 5 \beta_{1} q^{73} -3 \beta_{2} q^{75} -2 \beta_{3} q^{79} + 19 q^{81} + 3 \beta_{2} q^{83} + 12 q^{85} + 6 \beta_{2} q^{87} -7 \beta_{1} q^{89} + 20 q^{93} + 2 \beta_{3} q^{95} -3 \beta_{1} q^{97} -7 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 28q^{9} + O(q^{10})$$ $$4q + 28q^{9} + 12q^{25} - 24q^{29} + 8q^{37} + 24q^{53} + 40q^{57} + 76q^{81} + 48q^{85} + 80q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 4 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 8 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} + 4 \beta_{1}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.28825 0.874032 −0.874032 −2.28825
0 −3.16228 0 −2.82843 0 0 0 7.00000 0
1.2 0 −3.16228 0 2.82843 0 0 0 7.00000 0
1.3 0 3.16228 0 −2.82843 0 0 0 7.00000 0
1.4 0 3.16228 0 2.82843 0 0 0 7.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.a.x 4
4.b odd 2 1 inner 1568.2.a.x 4
7.b odd 2 1 inner 1568.2.a.x 4
7.c even 3 2 1568.2.i.y 8
7.d odd 6 2 1568.2.i.y 8
8.b even 2 1 3136.2.a.bz 4
8.d odd 2 1 3136.2.a.bz 4
28.d even 2 1 inner 1568.2.a.x 4
28.f even 6 2 1568.2.i.y 8
28.g odd 6 2 1568.2.i.y 8
56.e even 2 1 3136.2.a.bz 4
56.h odd 2 1 3136.2.a.bz 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.2.a.x 4 1.a even 1 1 trivial
1568.2.a.x 4 4.b odd 2 1 inner
1568.2.a.x 4 7.b odd 2 1 inner
1568.2.a.x 4 28.d even 2 1 inner
1568.2.i.y 8 7.c even 3 2
1568.2.i.y 8 7.d odd 6 2
1568.2.i.y 8 28.f even 6 2
1568.2.i.y 8 28.g odd 6 2
3136.2.a.bz 4 8.b even 2 1
3136.2.a.bz 4 8.d odd 2 1
3136.2.a.bz 4 56.e even 2 1
3136.2.a.bz 4 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1568))$$:

 $$T_{3}^{2} - 10$$ $$T_{5}^{2} - 8$$ $$T_{11}^{2} - 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -10 + T^{2} )^{2}$$
$5$ $$( -8 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( -20 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( -18 + T^{2} )^{2}$$
$19$ $$( -10 + T^{2} )^{2}$$
$23$ $$( -80 + T^{2} )^{2}$$
$29$ $$( 6 + T )^{4}$$
$31$ $$( -40 + T^{2} )^{2}$$
$37$ $$( -2 + T )^{4}$$
$41$ $$( -2 + T^{2} )^{2}$$
$43$ $$( -20 + T^{2} )^{2}$$
$47$ $$( -40 + T^{2} )^{2}$$
$53$ $$( -6 + T )^{4}$$
$59$ $$( -10 + T^{2} )^{2}$$
$61$ $$( -72 + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$( -80 + T^{2} )^{2}$$
$73$ $$( -50 + T^{2} )^{2}$$
$79$ $$( -80 + T^{2} )^{2}$$
$83$ $$( -90 + T^{2} )^{2}$$
$89$ $$( -98 + T^{2} )^{2}$$
$97$ $$( -18 + T^{2} )^{2}$$